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SUST Journal of Science and Technology (SUST JST)

Modification of Heisenberg’s Uncertainty Relation for a Damped Harmonic Oscillator under the Generalized Uncertainty Principle

Partho Karmaker, S. B. Faruque

Published Aug 16, 2026• 0 citations
Publisher

Abstract

We investigate the effects of the Generalized Uncertainty Principle (GUP), which intro- duces a fundamental minimal observable length, on the quantum dynamics of a damped harmonic oscillator. By incorporating a modified Heisenberg algebra into the framework, we derive the altered uncertainty relations for damped quantum systems and examine their dependence on the discrete energy spectrum. Our analysis reveals significant devia- tions from the standard Heisenberg uncertainty relation, demonstrating that the presence of a minimal length scale fundamentally modifies the quantum-classical correspondence in dissipative systems. We present explicit expressions for the modified uncertainty relations and discuss their physical implications for quantum measurements in damped oscillatory systems. These results contribute to our understanding of quantum mechanics at length scales approaching the Planck regime and provide testable predictions for experimental verification of GUP effects in dissipative quantum systems.

DOI
10.63512/sustjst.2025.1002

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We investigate the effects of the Generalized Uncertainty Principle (GUP), which intro- duces a fundamental minimal observable length, on the quantum dynamics of a damped harmonic oscillator. By incorporating a modified Heisenberg algebra into the framework, we derive the altered uncertainty relations for damped quantum systems and examine their dependence on the discrete energy spectrum. Our analysis reveals significant devia- tions from the standard Heisenberg uncertainty relation, demonstrating that the presence of a minimal length scale fundamentally modifies the quantum-classical correspondence in dissipative systems. We present explicit expressions for the modified uncertainty relations and discuss their physical implications for quantum measurements in damped oscillatory systems. These results contribute to our understanding of quantum mechanics at length scales approaching the Planck regime and provide testable predictions for experimental verification of GUP effects in dissipative quantum systems.
The generalized uncertainty principle (GUP) is a modification of Heisenberg's Principle predicted by several theories of quantum gravity. It consists of a modified commutator between the position and momentum. In this work, we compute potentially observable effects that GUP implies for the harmonic oscillator, coherent, and squeezed states in quantum mechanics. In particular, we rigorously analyze the GUP-perturbed harmonic oscillator Hamiltonian, defining new operators that act as ladder operators on the perturbed states. We use these operators to define the new coherent and squeezed states. We comment on potential applications.
Different proposed candidates of quantum gravity, such as string theory, noncommutative geometry, loop quantum gravity, and doubly special relativity, all predict the existence of a minimum observable length and/or a maximum observable momentum. These natural cutoffs modify the standard Heisenberg uncertainty principle leading to the so-called Generalized Uncertainty Principle (GUP). In this paper, we study the effects of minimal length and maximal momentum on the dynamics of harmonic oscillations. Within this framework we also study the status of Ehrenfest's theorem with the GUP.
in ordinary quantum mechanics for the position uncertainty since the minimum value for the uncertainty can be tend to zero. Different theories of quantum gravity, such as string theory, loop quantum gravity, and black hole physics, confirm a minimal observable length. Kempf et al have formulated The minimal length is a fundamental length for position measurement, although there is no lower bound a generalized uncertainty principle (GUP) by modifying the Heisenberg uncertainty principle to include an appearance of the length in quantum mechanics. In this study, we have found out the influence of the GUP on the particle for its confinement in a 1D potential box and calculated energy eigenvalues. The modified Schrödinger equation for the particle is factorized to be of second order to obtain the eigenvalues in a handy way. The obtained energy levels are modified from the usual result with GUP parameter dependency. The modification in the energy spectrum due to the GUP characterized by the presence of the minimal length has been also explored graphically.
Several approaches to quantum gravity suggest the emergence of a fundamental minimal length at the Planck scale. In quantum mechanics, this feature is naturally encoded through deformations of the Heisenberg algebra, leading to the Generalized Uncertainty Principle (GUP). While the phenomenological implications of GUP have been extensively explored, a consistent characterization of coherent states in minimal-length quantum mechanics remains elusive. In this work, we present a systematic analysis of coherent states for the one-dimensional harmonic oscillator. We show that the canonical equivalence among their standard characterizations - as eigenstates of the annihilation operator, displaced vacuum states and minimum-uncertainty wave packets - is generically lost in the presence of a minimal length. We then investigate the dynamical and semiclassical consequences of this inequivalence by comparing the evolution of generalized coherent states with that of states saturating the GUP. In particular, we demonstrate that minimal-length effects induce nontrivial deformations of phase-space trajectories and give rise to an intrinsic squeezing mechanism with no counterpart in ordinary quantum mechanics. These results establish a unified framework for coherence in GUP-based quantum theories and identify distinctive semiclassical signatures of minimal-length physics, opening a new avenue for probing quantum-gravitational effects.
We present the coherent states of the harmonic oscillator in the framework of the generalized (gravitational) uncertainty principle (GUP). This form of GUP is consistent with various theories of quantum gravity such as string theory, loop quantum gravity and black-hole physics and implies a minimal measurable length. Using a recently proposed formally self-adjoint representation, we find the GUP-corrected Hamiltonian as a generator of the generalized Heisenberg algebra. Then following Klauder's approach, we construct exact coherent states and obtain the corresponding normalization coefficients, weight functions and probability distributions. We find the entropy of the system and show that it decreases in the presence of the minimal length. These results could shed light on possible detectable Planck-scale effects within recent experimental tests.
Harmonic oscillator is significantly important in quantum gravity for its dynamical properties. Dynamics of the harmonic oscillator can be modified by inserting a modified uncertainty relation considered in quantum cosmology models such as string theory and quantum loop gravity theory. In this paper, the equation of damped quantum harmonic oscillator has been solved with respect to the modified uncertainty relation and the effects of this modification have been probed in the case of the damped oscillating system.